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Journal of Economic Behavior & OrganizationVol. 55 (2004) 1–23

Persistent parochialism: trust and exclusionin ethnic networks

Samuel Bowlesa, Herbert Gintisb,∗a Santa Fe Institute, University of Siena, Italy

b Santa Fe Institute, Columbia University, USA

Received 7 January 2003; accepted 2 June 2003

Available online 28 May 2004

Abstract

Decentralized groups such as close knit residential neighborhoods and ethnically linked busi-nesses often achieve high levels of cooperation while engaging in exclusionary practices that wecall parochialism. We investigate the contribution of within-group cultural affinity to the abilityof parochial groups to cooperate in social dilemmas. We analyze parochial networks in which thelosses incurred by not trading with outsiders are offset by an enhanced ability to enforce informalcontracts by fostering trust among insiders. We show that there is a range of degrees of parochialismfor which parochial networks can coexist with an anonymous market offering unrestricted tradingopportunities.© 2004 Elsevier B.V. All rights reserved.

PACS:C7; D2

Keywords:Parochialism; Exchange networks; Insider–outsider relationships

1. Introduction

Decentralized groups such as close knit residential neighborhoods and ethnically linkedbusinesses often achieve high levels of cooperation while also engaging in exclusionarypractices that we callparochialism. We investigate the contribution of within-group cul-tural affinity to the ability of parochial groups to cooperate in social dilemmas. Consistentwith some recent experimental evidence, cooperation among members of culturally ho-mogeneous groups does not come about because cultural affinity overrides individual self

∗ Corresponding author. Tel.:+1-413-586-7756; fax:+1-206-984-9873.E-mail addresses:[email protected] (S. Bowles), [email protected] (H. Gintis).URLs:http://www.santafe.edu/∼bowles, http://www-unix.oit.umass.edu/∼gintis.

0167-2681/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.jebo.2003.06.005

2 S. Bowles, H. Gintis / J. of Economic Behavior & Org. 55 (2004) 1–23

interest, thus leading members to behave altruistically towards others in their group. Rather,cultural affinity supports cooperation by altering the information structure of the interaction.We model the economics ofethnic networks, defined as sets of agents unified by similarityof one or more ascriptive characteristics engaged in non-anonymous interactions structuredby high entry and exit costs, but lacking a centralized authority.1

Networks manage such common pool resources as fisheries, irrigation, and pasturage(Acheson, 1988; Wade, 1988; Ostrom, 1990), regulate work effort and risk sharing inproducer cooperatives (Whyte, 1955; Homans, 1961; Lawler, 1973; Craig and Pencavel,1992, 1995; Platteau and Seki, 2001), enforce non-collateralized credit contracts (Udry,1993; Banerjee et al., 1994) promote neighborhood amenities in residential communities(Sampson et al., 1997), and privately enforce contracts among traders in securities (Baker,1984) and diamond markets in the US (Bernstein, 1992), and food markets in Madagascar(Fafchamps and Minten, 2001).

Networks often do quite well economically, as the flourishing informal ethnic businesslinkages among new immigrants to the United States and the United Kingdom attest (Rauch,1996; Granovetter, 1985; Kotkin, 1993). For instance, Cambodians run more than 80 percent of California’s doughnut shops, raising funds from friends, family, and ethnic creditassociations (Kaufman, 1995). Similarly, Indians own more than a third of the motels inthe United States, frequently raising initial capital through unsecured loans from extendedfamily members (Woodyard, 1995).

Among the problem-solving capacities of networks are the powerful contractual enforce-ment mechanisms made possible by small-scale interactions, notably effective punishingof those who fail to keep promises, facilitated by close social ties, frequent and variegatedinteractions, and the availability of low cost information concerning one’s trading partners.This problem-solving capacity allows networks to counteract the restricted gains from tradeand foregone economies of scale due to small size and exclusionary practices.2 Members,of course, do not normally express their identification with networks in terms of their eco-nomic advantages. Rather, they typically invoke religious faith, ethnic purity, or personalloyalty. These sentiments often support exclusion or shunning of outsiders. We model thesepractices, which we term parochialism, inSection 2.

1 The theory of social exchange, initiated in sociology byBlau (1964)andHomans (1958), and in anthropologybySahlins (1972)provide insights into the economics of networks. For contributions by economists, seeBen-Porath(1980), Hollander (1990), Iannaccone (1992), Kandori (1992), Wintrobe (1995), Greif (1994), Akerlof (1997),Pagano (1995), Benabou (1996), Durlauf (1996), Kranton (1996), Taylor (1997), andGlaeser (2000).

2 The advantages of trade with those deemed “outsiders” is a common explanation of the permeability of networkboundaries in small-scale societies (Adams, 1974) and of the extinction of very restrictive networks in favor ofmore inclusive entities (Gellner, 1985; Weber, 1976). A particularly well-documented example of this tensionis Greif’s account of how the competitive advantages stemming from the superior within-network contractualenforcement capabilities of the tight-knit 13th century community of Maghribi merchants was eventually offset bytheir reduced ability to engage in successful exchange with outsiders, resulting in their inability to compete withthe more individualistic Genovese traders. Ben Porath (p. 14) develops similar reasoning concerning the economiccapabilities of families and other face to face networks:

The transactional advantages of the family cannot compensate for the fact that within its confines the returnsfrom impersonal exchange and the division of labor are not fully realizable.

S. Bowles, H. Gintis / J. of Economic Behavior & Org. 55 (2004) 1–23 3

Networks arise in part because people choose to associate with others who are similar tothemselves in some salient respect (Lazarsfeld and Merton, 1954; Thibaut and Kelly, 1959;Homans, 1961). Among the salient characteristics on which this choice operates are raceand ethnic identification, and religion (Berscheid and Walster, 1969; Cohen, 1977; Kandel,1978; Tajfel et al., 1971; Obot, 1988). Conversely, people often seek to avoid interactionswith those who are different from themselves.

We seek to illuminate the following puzzle: why do parochial sentiments and practices,often identified with archaic social distinctions and intolerance of strangers, persist in mod-ern market-based and liberal societies? Our response, briefly, will be that parochialism isnot an anachronistic remnant of the past, perpetuated by inertia, but rather that networksbased on parochialism solve economic problems that are resistant to market- or state-basedsolutions. Persistent parochialism is thus explained at least in part by the problem solvingcapacities of the network interactions that parochialism underpins.

When interactions among group members are characterized by material payoffs that takethe form of prisoner’s dilemma, public goods or other social dilemmas, parochialism maycontribute to successful cooperation in three ways. First, members may feel more altru-istic toward the ethnically similar members of their group than towards “outsiders”. It iswell known that if each member values the payoffs gained by other members sufficientlyhighly, mutual defect is no longer the dominant strategy equilibrium of these interactions,and mutual cooperation may be a stable Nash equilibrium. The “minimal group” experi-ments initiated by Tajfel et al. and his colleagues in the early 1970s are often interpretedin this way. Experimental subjects were assigned to groups on the basis of some trivialdistinction (commonly their preference for paintings by Paul Klee over those of WassilyKandinsky). In-group favoring behavior was quite pronounced in these experiments. Laterprisoner’s dilemma and common pool resource experiments found higher levels of coop-eration when the players are members of the same minimal group than when they are notmembers of the same group (Kramer and Brewer, 1984). However, a series of recent exper-iments byYamagishi (2003)and his associates show that experimental subjects’ allocationsfavor in-group members not because of altruistic sentiments towards those who are sim-ilar to themselves, but because they expect reciprocation from in-groupers and not fromout-groupers.3

A second way that cultural affinity could support cooperation is by enhancing the forceof the forms of altruistic punishment that often sustain cooperation in experimental pub-lic goods games (Barr, 2001; Fehr and Gächter, 2002; Masclet et al., 2003). For ex-ample, it is plausible that the shame induced by criticism for selfish behavior from afellow group member is greater when the critic shares the defector’s beliefs about goodbehavior.

The third reason parochial groups may cooperate in interactions that would take the formof prisoner’s dilemmas or public goods games in non-parochial settings, unlike the firsttwo reasons, does not concern the members’ preferences. Rather it is the effect of culturalaffinity on the information structure of the interaction, allowing equilibrium strategies un-available in the information environments of less parochial groups. Our model explores

3 Yamagishi’s work does not show that within-group altruism does not exist. Rather, he shows that the minimalgroup experiments provide no evidence to this effect.


this mechanism as a contributor to the success of parochial groups in addressing socialdilemmas.

We do not suggest, of course, that the contribution of parochial sentiments and practicesto economic performance of groups is the sole reason for their persistence. Ethnic, racialand other group identities arise and persist for a multitude of reasons, many of them farless benign than those studied here.Loury (2001)provides a compelling account of someof these reasons.

A successful model of parochialism in networks must satisfy four conditions. First, itshould be a general equilibrium model in which the size of the anonymous market andthe size and array of parochial networks are endogenously and simultaneously determined.We do this by constructing a model with explicit network–market interactions. Second, therange of ascriptive traits that can be the basis for a successful parochial network must beendogenously determined. We do this by developing the concept ofascriptive filters, each ofwhich is associated in equilibrium with a network of a particular size and composition. Third,decision-making and information exchange in networks should be completely decentralized,thus distinguishing networks from teams with a centralized decision-making structure andgroups all of whose members have access to a common pool of information. Finally, theexchanges that take place between group members must be one-shot, rather than beingbased on repeated dyadic interactions in which cooperation is ensured by superior gains oflong-term interaction. This is not because most interactions within networks are one-shotas an empirical matter, but rather in order to illuminate what is distinctive about parochialnetworks. The major benefit of participating in a parochial network is the ability to gaininformation about a potential trade partner at low cost from other network members. Dyadicinteractions that are sufficiently repeated operate effectively without such information. Ourmodel meets all these requirements.

Our model is closest in spirit toMcElreath et al. (2003), who show that ethnic mark-ers are likely to be salient if they correlate with shared beliefs that facilitate cooperation.Like McElreath et al. (2003), ethnic affinity in our model does not support cooperationby means of positive sentiments among group members, but rather by removing barri-ers to exchange in the presence of incomplete contracts. Unlike their paper, however, weconsider the distribution of a population between ethnically marked groups and an un-marked pool, and we model the reasons why shared ascriptive traits contribute to suc-cessful exchange, while also precluding some gains from trade. Our paper shares withthe general equilibrium framework of interaction between a system of anonymous marketexchange and another system of reciprocal relationships in small groups, as well as thenotion that the comparative advantage of the anonymous market is in product diversity,while that of small groups is contract enforcement. However, Kranton’s group relation-ships are dyadic interactions that discourage free-riding through repetition and the threatof withdrawal from the relationship, while in our model, all trades are one-shot events,free-riding being discouraged by a decentralized network information structure that es-tablishes the reputation of group members. Like Greif, we consider the coexistence ofalternative trading strategies. However, unlike Greif, we assume entirely decentralized in-formation flows, the quality of which is affected by the degree of parochialism of the net-work. Cultural affinity of group members per se plays no role in either Greif’s or Kranton’smodel.


The mechanism for the success of networks explored in this paper is their ability topromotetrust.4 We consider a large population of agents who, while economically identical,are distinguishable by markers indicating group membership. These agents take three typesof actions. First, they locate in one of a variable number of networks or remain outsideany network in what we will call the ‘anonymous pool’ of traders. Second, they choosestrategies that govern their behavior with trading partners. Third, they update these strategiesin light of their relative payoff compared to other available trading strategies. We explorethe evolution and equilibrium frequency of behaviors within networks, the distribution ofpopulation between networks and the anonymous pool, and the size and number of networks,under the influence of parochial practices. We conclude with a series of implications of themodel concerning the impact of the evolving information structure of modern economieson the likely future importance of parochial networks.

2. Parochialism and heterogeneity in networks

Individuals implement their desires to associate with others like themselves by engag-ing in what we termparochial practices. These practices take the form of refusal to tradewith ‘outsiders’ that,ceteris paribus, lower the returns to members of parochial networks.McMillan and Woodruff (1999, p. 1285)study of trust among businesses in Vietnam sug-gests the salience of this tradeoff:

Trading relations in Vietnam’s emerging private sector are shaped by two market fric-tions: the difficulty of locating trading partners and the absence of legal enforcement ofcontracts. Examining relational contracting, we find that a firm trusts its customer enoughto offer credit when the customer finds it hard to locate an alternative supplier. . . . Cus-tomers identified through business networks receive more credit. These network effectsare enduring, suggesting that networks are used to sanction defaulting customers.

Thus, in some cases, homogeneity may offer advantages offsetting the foregone gainsfrom trade. Parochial communities such as the Pennsylvania Amish and the Canadian Hut-terites have expanded their numbers and thrived economically.5 Among the Amish, forexample, distinctive dress, dialect, and technology construct a “cultural moat” around thegroup and, acting as “armaments of defense, they draw boundary lines between church

4 Our model develops insights provided by a number of contributions to the sociology of networks. Granovetterwrites (pp. 490–491)

. . . social relations, rather than institutionalized arrangements or generalized morality are mainly responsiblefor the production of trust in economic life.

For additional ways in which networks solve coordination problems stemming from incomplete contracts, seeBowles and Gintis (1998, 2002).

5 SeeWilson and Sober (1994)andKraybill (1989). Hechter (1990)found that two indicators of group homo-geneity, common ethnic background and uniform style of dress, were among the few robust predictors of survivalof utopian communes established in the late 18th and early 19th century in the United States. He interprets thisfinding as in part reflecting variable information costs. See alsoLonghofer (1996)for a model of the relationshipbetween cultural affinity and monitoring costs.


and world [to] announce Amish identity to insider and outsider alike” (Kraybill 50,68).Yet the boundaries erected around Amish culture have not prevented economic success andpopulation growth. Further, the record of successful ethnic business affiliations suggeststhat parochialism may not only foreclose opportunities, but also contribute to the successof networks.

We model parochialism as an ascriptive on given traits of those with whom one mightinteract, a particular form of parochialism excluding those with ‘objectionable’ traits.6

Individuals who do not exclude those with objectionable traits are themselves objectionable,even if their traits per se are not objectionable.7 Thus any ascriptive different from one’sown is assumed to be objectionable, so networks will made up of individuals with the sametype of parochialism.

It might be thought that this assumption presents an incentive problem for memberswhose traits conform to the filter but who stand to gain by secretly transacting with agentswhose traits do not conform. This is not the case. To see this, suppose network memberstrade only among themselves. Then, an agent who considered trading with a non-memberwould know that he could obtain no information concerning the non-member’s behavior.The non-member would have no incentive to act cooperatively; there would be no reputationvalue of doing so because he does not trade with other network members. Hence such atrade would have the payoffs of an anonymous market transaction, but if such a trade werebeneficial, the member would be better off in leaving the network entirely. As a result,however different they are in other respects (for example, pursuing different strategies ineconomic interactions or differing in a trait not covered by the filter), they will agree on thecommon traits for which their ascriptive filter selects.8

In principle, any individual trait, such as liking the color blue or eating lots of fish, couldbe the basis for eligibility for membership in a parochial network. Some characteristics,including such ascriptive traits as race, ethnicity, and native language, however, are muchmore conducive to network formation. First, such ascriptive characteristics are difficult tofake or acquire, thus solving the problem of limiting immigration into the network when it issuccessful. Second, individuals with shared ascriptive traits are likely to have shared values,means of communication, beliefs, and tastes. Hence, two members of an ethnic network aremore likely to agree on and communicate what counts as acceptable trading behavior of athird party than are two randomly chosen members of the population who happen to agreeon a trivial and easily changed characteristic.

6 Iannaccone analyzes a more active form of parochialism, in which membership in a network subject to par-ticipatory crowding is restricted to those who are willing to accept “stigma, self-sacrifice, and bizarre behavioralrestrictions”.

7 Lazarsfeld and Merton term this second order exclusiveness “value homophily” and present evidence for itwith respect to racial attitudes: white ‘racial liberals’ prefer not to associate with white ‘racial illiberals’ andconversely.Sugden (1986)showed this to be a coherent basis for group formation, using the concept of being“in good standing” in the group. An individual who lacks one or more of the group’s core characteristics is notin good standing. An individual who associates with an individual who is not in good standing is himself not ingood standing. Otherwise, an individual is in good standing. A network consists of all individuals who are in goodstanding with respect to its core characteristics.

8 The assumption that ethnic network members share a common filter is justified empirically, but relaxing itwould not qualitatively alter our analysis, while complicating it considerably. To the extent that network membershave disparate filters, information exchange will be more costly and less efficient.


Suppose in pairwise strategic interactions, agents can condition their actions on whetherthe other player is an ‘insider’ or an ‘outsider’. Each individual has a certain set of traits(ethnicity, language, physical attributes, cultural or demographic characteristics, and thelike), which we take to be fixed. We label these traitsj = 1, . . . , n, each individual beingcharacterized by a trait profilea = a1, . . . , an, where eachaj = 1 oraj = 0 according asto whether the individual does or does not possess traitj. LetA be the set of all possible traitprofiles. An individual with traitsa ∈ A may have a ‘ascriptive filter’, defined as a vectorb ∈ A such thatb ≤ a, in the sense that the individual with traitsa also has all the traitsindicated byb.

Let us define an individual asb-parochialif he has all theb-traits, and he trades only withotherb-parochial individuals. We also refer tob-parochial agents asinsiders(the trait vectorb being assumed), and we refer to a non-insider as anoutsider. A outsider is therefore anindividual who lacks one or more of theb-traits, who trades with someone who lacks one ormore of these traits, or who trades with someone who trades with someone who lacks oneor more of these traits, and so on. In effect,b-parochial agents choose a subset of the traitsthey possesses (the unit-entries inb that are also unit-entries ina) and consider as insidersexactly those agents who have these traits and are ‘like-minded’ in the sense that they havethe same criteria for distinguishing between insiders and outsiders. We assume throughoutthat the property of beingb-parochial is common knowledge.

This formalization reflects our view that the immense variety of noticeable individualdifferences and similarities is the raw material on which parochialism works. A particularb-parochialism makes some subset of these differences behaviorally salient while ignor-ing others. For instance, suppose the array of traits are (‘female’, ‘French-speaking’). Anagent with characteristicsa = 11 is a female Francophone. Such an individual could beb-parochial forb = 11 (insiders are like-minded female Francophones),b = 01 (insidersare like-minded Francophones),b = 10 (insiders are like-minded females), orb = 00(insiders are like-minded i.e., they treat all others as insiders).

It is clear from this example that individuals may differ in the extent of their parochialism.As we will see below, these differences will affect both the size and heterogeneity ofnetworks, which in turn will influence the gains from within-network trading, but first weneed to formalize thedegree of parochialismof a network and theexpected communicationdifficultywithin a network.

Suppose, for example, there are three salient binary traits, language, nationality, and“race”. We define trait profiles so that a person with the trait profilea = 111 is French-speaking, European, and “white”, while a 000 is non-French-speaking, non-European, andnon-white. A 010 person is a non-French-speaking, European non-white, and so on, coveringall eight possible trait types generated by these categories. Thedegree of parochialism,ρ, is a measure of the stringency of an individual’s filter, as indicated by the minimumnumber of ways another must resemble the individual in order to be considered an insider.Thusρ = 0 indicates the total absence of parochialism, whileρ = 3 indicates completeparochialism (for the three-trait case), meaning that such individuals will associate onlywith those identical to them in all three traits. Because networks will be homogeneous withrespect to the degree of parochialism, we can speak ofρ as a network trait.

We cannot in general compare the degree of parochialism between arbitrary filters. Wecan, however, determine which of two filters that differ only in one entry is more parochial.


For instance, we cannot say that excluding Jews or excluding blacks is more parochial, butwe can say that excluding Jews and women is more parochial than simply excluding Jews.By extension, we can compare two filters if their differences can be expressed by a seriesof such comparisons. In other words, ascriptive filterspartially order the setP of parochialnetworks.9 Given networksN ∈ P consisting ofb(N)-parochial agents andM ∈ P withb(M)-parochial agents, we sayM is more parochialthanN if b(M) > b(N), so everymember ofM would be admitted toN.

To explore the impact of the degree of parochialism on the ability of network membersto cooperate effectively, we assume that communication difficulty rises with the number oftrait differences. Letµij represent thecommunication difficultybetween individuals of traittypesi andj, defined as the number of traits on which they differ. In the three trait case, forinstanceµij can take on the values 0, 1, 2, and 3.

Now consider the communication difficulty arising among randomly paired individualsin networks with a given ascriptive filter. Suppose there aremtypes of agents in the network,where the frequency of typei ispi ≥ 0, i = 1, . . . , m. Assuming random pairing of agents,theexpected communication difficultyis then given by

µ =m∑

i,j=1

pipjµij .

For instance, continuing our previous example, assume that each of the eight trait typesis equally common in the larger population from which the networks are drawn and thatthe relative frequency of types who are admitted to a network is equal to their relativefrequency in the larger population. Then ifρ = 3, there are no communication difficulties,as all members of the network have the same three traits, soµ = 0. If ρ = 2, by contrast, thenetwork will be composed of equal numbers of two types similar with respect to two traitsand different with respect to the other trait. Thus a random pairing will yield pairs witha one-trait difference approximately half the time, yielding an expected communicationdifficulty µ = 1/2. By similar reasoningρ = 1 yieldsµ = 1 andρ = 0 givesµ = 3/2.This reasoning is readily generalized to larger numbers of traits and trait groups of unequalsize.

We can show thatµ is decreasing inρ on any totally ordered subset of the setP ofparochial networks. Consider a networkN ∈ P, and suppose there arem types inN, typeioccurring with frequencypi. In addition, consider the more parochial networkM gotten byreplacing members ofN who lack a certain previously ignored trait, say trait 1, with agentswho possess this trait and have otherwise identical trait profiles as the agents they replace.Then, provided that the fraction of agents inNwith trait 1 lies strictly between zero and one,members of the newly constituted, more homogeneous, networkM will enjoy strictly lesscommunication difficulty. To see this, note that before the change an agent had a positivechange of meeting an agent with a different value for trait 1, and now has a zero chance.Moreover, no other meeting probability has changed, so communications costs must fallfor all agents. We have the following theorem.

9 Formally, a partial ordering< on a setSis a transitive binary relation onS, and a total ordering onSis a partialordering such that, for any two elementsA, B ∈ S, eitherA = B, A < B, orB < A.


Theorem 1 (Parochialism and communication difficulty).Increasing parochialismin any totally ordered subset of the set of parochial networksP reduces communicationdifficultyµ.

3. The costs and benefits of networks

In this section, we analyze the effect of the degree of parochialism on the information andtrading opportunities available to members of a single network, taking as given the com-position of and payoffs to members in other networks and the anonymous pool of traders.Theorem 1shows that level of expected communication difficultyµ(ρ) is decreasing inρon any totally ordered subsetPo ⊂ P. We shall assume network sizex(ρ) is decreasingin ρ on Po, for the obvious reason that increased parochialism reduces the pool of po-tential migrants to the network. We will later examine the manner in which the sizex(ρ)of a single network depends on the composition of other networks and the anonymouspool.

Suppose members of any networkN ∈ Po are eithertrustworthyor untrustworthy(wemodel trustworthiness inSection 4). We will call the trustworthy memberscooperatorsand the untrustworthy membersdefectors. We will show that for agents trading withina network the quality of the signalp(ρ) is an increasing function ofρ on Po, and theprobability q(ρ) of meeting a partner for mutually beneficial trade is a decreasing func-tion of ρ onPo. Signal qualityp(ρ) is increasing inρ onPo for two reasons. First, moreparochial networks are smaller, and smaller networks possess more information concern-ing each individual. Second, more parochial networks have lower communication costs,leading to a higher likelihood of correctly ascertaining the trustworthiness of potentialtrading partners. Similarly,q(ρ) is decreasing inρ on Po because more parochial net-works have fewer members, and hence each member faces a lower probability of meet-ing a potentially mutually beneficial trading partner. Moreover, a more homogeneousset of agents is less likely to enjoy complementary patterns of excess supply anddemand.

For any totally ordered set of networksPo ⊂ P, we define anetwork information structureI (x(ρ),κ,po(ρ)) with the following properties. Each member of a network ofx(ρ) individualsknows the type (cooperator/defector) ofκ other members. An individual who seeks to knownthe type of a specific memberj of the network receives informant messages randomly frommembers of the network, until a message arrives from an informant who knowsj’s type. Theinformant’s report ofj’s type is correctly communicated with probabilityτ, which variesinversely withµ(ρ).10

We can then express the probability that the individual receives the correct information,p(ρ) as follows. Letq be the probability of receiving correct information if the agent doesnot know his partner. Then

10 Note that an incorrect communication can occur either because the informant errs, or because the informantlies. There is no gain to lying in our model, so it is permissible to treat willful misinformation as a stochastic eventthat does not vary systematically with the variables and parameters of our model.


p = κ

x+(1 − κ

x

)q, (1)

q = κ

xτ +

(1 − κ

x

)q. (2)

Eqs. (1) and (2)have the following interpretation. With probabilityκ/x personj is known tothe individual, but with probability (1−κ/x) the individual, not knowingj personally, mustconsult an informant. The informant will knowj with probabilityk/x and will communicatethis successfully to the individual with probabilityτ. However with probability (1−κ/x) theinformant will not knowj, and the individual must seek another informant, yielding therecursion expressed above.

Eqs. (1) and (2)may be solved as

p(ρ) = κ

x(ρ)+(

1 − κ

x(ρ)

)τ(ρ). (3)

Clearlyp(ρ) is increasing inρ, sincex(ρ) is decreasing andτ(ρ) is increasing. Thereforewe have the following theorem.

Theorem 2 (Parochialism and signal quality).Consider a totally ordered subsetPo ⊂ Pof parochial networks, and let I (x(ρ), κ, τ(ρ)) be the information structure of a networkin Po. Then the average signal quality p(ρ) onN is an increasing function of the level ofparochialism.

We model this as a problem of finding partners with whom to exchange goods or services,but an equivalent formulation would model the problem of finding advantageous matchingsfor some joint production activity with skill complementarities between demographic groupsgiving an advantage to more heterogeneous group. To specify the shape ofq(ρ) onPo ⊂ P,suppose agents produce goods for trade in the morning and take them to market for trade inthe afternoon. Goods are perishable and cannot be stored. Suppose there arex(ρ) agents inthe network and there are goods 1, . . . , k, corresponding to which there are ‘marketplaces’that have exogenously given relative sizesf1, . . . , fk (

∑i fi = 1). Marketplacei thus has

absolute sizexi = fix(ρ) for i = 1, . . . , k. The members who are to compose thisxi areassigned randomly at the start of the trading period. Each agent decides to be a buyer or aseller that period. Buyers and sellers in the same marketplace are randomly paired, and ifthe number of buyers and sellers differ, a random selection of agents will make no trade atall, and as a result trades on the anonymous market, receiving a payoff normalized to zero.Suppose the distribution of individual capacities and preferences differ among groups, soa network composed of many groups will have a greater variance of both preferences andproduction possibilities than a homogeneous group. To capture the effect of heterogeneityon the probability of trade, we assume that agents of the same type are more likely to belocated on the same side of the market. Thus the expected fractionψi(ρ) of agents on thedemand side of marketplacei will be distant from 1/2 when networks are very homogeneousand close to 1/2 when networks are heterogeneous.

The more parochial a network is, the less likely will agents be able to make a trade, fortwo reasons. First, the more parochial networks will be more homogeneous, so bunchingmany agents on one side of the market will happen frequently. Second, even if the expected


number of buyers and sellers were equal in every market, the smaller each market is, themore likely mismatches will be on any particular market day.Theorem 3, reflecting thislogic, is proved inAppendix A.

Theorem 3 (Gains to network heterogeneity).Let q(ρ) be the probability of making a tradewhen network parochialism isρ. Then q(ρ) is decreasing inρ.

4. Trust in networks

To model the population of traders, consider a gameG where many agents are randomlypaired to play a one-shot prisoner’s dilemma in which each receivesc if they both defect,each receivesb if they both cooperate, and a defector receivesa when playing against acooperator, who receivesd. The assumptions of the prisoner’s dilemma then requirea >

b > c > d and 2b > a + d (the latter inequality ensuring that mutual cooperation yieldshigher average payoffs than defect/cooperate pairs). The coordination failure underpinningthe prisoner’s dilemma structure of this interaction arises because some aspects of the goodsor services being exchanged are not subject to costlessly enforceable contracts. The defectstrategy, for example could represent supplying shoddy goods where product quality is notsubject to contract.

We assume that any agent can trade in the anonymous market, the payoff to which wenormalize to zero. As agents in this market are unknown to one another, their interactionsare effectively non-repeated, precluding the kinds of informal contractual enforcement thatmay be possible for interactions within networks. It is reasonable to suppose that the kindsof goods and services traded in the anonymous market will tend to be those for whichrelatively complete and easily enforceable contracts can be written. Networks, by contrast,may specialize in the exchange of more difficult to contract goods and services.

An agent in a network can refuse to trade with his current partner, in which case weassume he trades on the anonymous market instead. If an agent does trade within thenetwork, the payoffs to mutual cooperation exceed the payoffs available in the anonymousmarket (b > 0) but at the same time the payoffs to mutual defection are inferior to thepayoffs of trades in the anonymous market (c < 0). This assumption is based on the notionthat with incomplete contracting, trading agents expose themselves to a greater level ofharm than would be the case with complete contracting. Thus operating within a networkis disadvantageous compared with operating in the anonymous market, unless the level ofcooperation within the network is sufficiently high.

We assume that each agent precommits to following one of three available ‘norms’. Thefirst, which we callDefect, is to defect unconditionally against all partners. The second,which we callTrust, is to cooperate unconditionally with all partners. The third, whichwe call Inspect, is to monitor an imperfect signal based on information provided by othermembers of the network indicating whether or not one’s current partner defects againstcooperators. We assume the signal correctly identifies a Defector with probabilityp andcorrectly identifies a non-Defector with the same probabilityp. The Inspector then refusesto trade with a partner who is signaled as a Defector and otherwise plays the cooperatestrategy. Thus when either partner to a within-network exchange refuses to trade, each


Fig. 1. The Inspect–Trust–Defect game.

receives payoff 0 which, according to the reasoning of the previous paragraph, is betterthan the mutual defect payoffc < 0.11 We assume that the signal is costlessly observed.Assuming a (not excessively large) positive cost of inspecting changes our results in anintuitively expected way, so we abstract from such costs in the interests of simplicity. Thepayoff matrix for a pair of agents has the normal form shown inFig. 1. We writeG(p) forthe game with signal accuracyp.

Let αt , βt , andδt be the fraction of the population playing Inspect, Trust, and Defect attime t, respectively. We assume these are continuous variables. Letπt

I , πtT, andπt

D be thepayoffs to the strategies Inspect, Trust, and Defect at timet, respectively, against the mixedstrategy given by (αt, βt, δt). We find that

πtI = bp(pαt + βt) + d(1 − p)δt, (4)

πtT = b(pαt + βt) + dδt, (5)

πtD = a(αt(1 − p) + βt) + cδt, (6)

πt = αtπtI + βtπ

tT + δtπ

tD, (7)

whereπt is the average payoff in the game. Equating the payoffs to the three pure strategies,we find that the Nash equilibrium frequencies (α∗, β∗, δ∗) satisfy

α∗ = −adp+ b(d(2p − 1) + c(1 − p))

D, (8)

β∗ = p(ad(1 − p) − b(d(2p − 1) + c(1 − p)))

D, (9)

δ∗ = ab(1 − p)(2p − 1)

D, (10)

where

D = a(b(1 − p)(2p − 1) − dp2) + b(1 − p)(d(2p − 1) + c(1 − p)).

We have the following theorem.

Theorem 4 (A trust equilibrium). There is ap∗ < 1 such that forp∗ < p < 1, G(p) has aunique Nash equilibrium(α∗(p), β∗(p), δ∗(p)). In this equilibrium all three types of players

11 It is easy to show that other actions available to an Inspector who receives a signal indicating a defectingpartner involve either mimicking the behavior of Trusters or Defectors or else are strictly dominated by playingas indicated above. We thus lose nothing by ignoring such alternatives.


occur as strictly positive fractions of the population. The payoffπ∗(p) in this equilibriumis positive and an increasing function of p, the fraction of Defectorsδ∗(p) is a decreasingfunction of p, and the fraction of Trusters is an increasing function of p.

To prove the theorem, define

p∗ = max

[1 − c

d,

a

2b

(√4b

a+ 1 − 1

)]. (11)

Note thatp∗ ∈ (0.618,1), sinced < c < 0 anda > b > 0. Then it is easy to show that

d(1 − p) > c and bp2 > a(1 − p) (12)

for all p such thatp∗ < p < 1. The inequalities in (11) imply that Inspect is a best responseto Defect, so mutual defect cannot be an equilibrium, and that Defect is not a best responseto Inspect, so a Defect–Inspect equilibrium is precluded. A routine check then indicates thatthere are no Nash equilibria involving fewer than all three strategies.12 Hence by Nash’sexistence theorem, there is an equilibrium ofG(p) involving all three strategies. This provesthatp∗ has the asserted property.Eqs. (8)–(9)imply

π∗ = −abd(2p − 1)2

D, (13)

soπ∗/δ∗ = −d(2p−1)/(1−p) > 0, showing that payoffs are positive. A tedious calculationverifies that

dπ∗

dp= δ∗ 2(−d)

b(d(2p − 1) + 2c(1 − p)) + a(b(2p − 1) − 2dp)

ab(1 − p)2(2p − 1).

The denominator in the fraction is positive and the numerator can be written as

2(b(a − c) − d(a − b))(p − 12) + bc− da,

which is clearly positive. To prove the final assertion, we calculate

dδ∗

dp= δ∗ 2ab(2p − 1)2(1 − p)2

bc(1 − p)2 + adp(3p − 2).

The denominator in this expression is less thanbc(2p − 1)2 < 0, from which the assertionfollows.

The intuition behindTheorem 4is simple. Consider the simplex

T = {(α, β)|α, β, α + β ∈ [0, 1]}.12 When the second inequality in (11) fails, but

p <(a/b) + (1 − (c/d))

1 + (a/b) + (1 − (c/d))

there also is no Nash equilibrium involving fewer than three pure strategies. We will ignore this alternative, forease of exposition.


Trust

Defect Inspect

T

δ

β

α

Fig. 2. A simplex phase diagram for G(p) when p∗ < p < 1. The frequency of Inspect, Trust, and Defect are α,β and δ, respectively. The trust equilibrium is at T. Note that there are no equilibria along the two-dimensionalboundary of the simplex, since each pure strategy can be invaded by another. The solid interior lines divide thesimplex into regions with differing dynamics. The line from T to the all Trust vertex is the locus of points suchthat dβ/dt = 0 while dδ/dt > 0 and dα/dt < 0, for example, so a group with a composition at a point on thislocus point will evolve in the direction given by the arrow.

By Nash’s existence theorem there is an equilibrium within T. However Trust is strictlydominated by Defect, and Inspect is strictly dominated by Trust (since Inspectors refusesome profitable trades, while Trusters do not). When the two inequalities (12) hold, Defectis also strictly dominated by Inspect. Therefore all Nash equilibria must be confined to theinterior of T. However, it is easy to check that there is only one possible candidate, whichthus exists and is unique. A phase diagram for the model is presented in Fig. 2.13

The replicator equations are then given by

dαt

dt= αt(π

tI − πt), (14)

dβt

dt= βt(π

tT − πt), (15)

reflecting our assumption that norms are implicated in the response to relative payoffs.We then have the following theorem.

Theorem 5 (Stability of the trust equilibrium). For p > p∗, the unique equilibriumP =(α∗, β∗, δ∗) of G(p) either is stable or has paths starting sufficiently near P that converge

13 We must also check on the dynamic properties of the interior Nash equilibrium. There is no guarantee that thisequilibrium is evolutionarily stable. Indeed, the reader can check that for a = 2, b = 1, c = −1 and d = −2, theequilibrium is not evolutionarily stable for p ≥ 0.78, while if we change a to a = 3, it is evolutionarily stable.However, evolutionary stability is a sufficient, though by no means necessary, condition for dynamic stability(Gintis, 2000, Chapter 10). Therefore, we must inspect a plausible dynamic, which we take to be the replicatordynamic (Taylor and Jonker, 1978).


to a periodic orbit of the replicator dynamic. In the latter case, the time averages of thepayoffs along the periodic orbit for the three strategy types are all equal toπ∗(p). Thusin either the stable or limit cycle case, the long-run expected payoff to an agent isπ∗(p),which is an increasing function of the signal quality p.

The first assertion follows directly from the Poincaré–Bendixson Theorem (Perko, 1991,p. 227), and the second from an ergodic theorem—Theorem 7.6.4 in Hofbauer andSigmund (1998, p. 79). By virtue of this theorem, we will therefore refer to either thestable or limit cycle case as a stable equilibriumof G(p).

It is easy to check that when p < p∗, there are only All Defect, or Defect/Inspectequilibria, both of which yield negative expected payoff. The first is stable and the secondunstable in the replicator dynamic. We assume the network disbands in such cases, so wetake π∗(p) = 0 for p < p∗.

5. The limits of sustainable parochialism

The characterization of the equilibrium given by Theorems 4 and 5 allow us now toconsider the effects of varying the degree of parochialism on the payoff to network members.

Consider the game G′(ρ), where ρ is the degree of network parochialism, differing fromG in two ways. First, the payoff to the prisoner’s dilemma stage game is the payoff in Gmultiplied by the decreasing function q(ρ) (see Section 3) minus a fixed cost c > 0 ofseeking a within-network transaction. Second, we assume the probability p of correctlyidentifying the type of a potential trade partner is an increasing function p = p(ρ) withinany totally ordered set Po ⊂ P of networks, as per Theorem 2. We call the game G′(ρ) thevariable parochialism networkgame.

We assume that there is a degree of parochialism ρmin on Po, such that p(ρ) > p∗ forρ > pmin. Thus there is a stable interior equilibrium for G′(ρ) for ρ > pmin. The equilibriumpayoff in G′(ρ), for ρ > pmin is then

π(ρ) = q(ρ)π∗(p(ρ)) − c (16)

and the equilibrium frequencies of Inspectors and Trusters can be written as α∗(p(ρ)) andβ∗(p(ρ)) on Po, respectively.

Theorem 6 says that if the number of agents of a particular type are either too smallor too large, this type cannot sustain a network equilibrium. This reflects the commonobservation that networks generally consist of minorities, but when insufficiently numerous,such minorities can do no better than operate within the anonymous pool because tradingopportunities become too rare to offset the transactions cost c of seeking a within-networktrade. Similarly, when the network becomes too large, signal quality becomes too low tosupport a trust equilibrium. We have the following theorem.

Theorem 6 (Equilibrium network size). LetPo ⊂ P be a totally ordered set of networks,so that each networkN ∈ Po is characterized by a particular parochialism levelρ. Forsufficiently small transactions costc, there is a non-empty interval(xmin, xmax) such that a


NetworkSize x

xmin xmax

q(r)π(x; ρ)

Equilibriumpayoff

c

Fig. 3. Payoffs in a trust equilibrium and network size for a network with a given level of parochialism. Note thatxmax is given by (17), and xmin is determined implicitly by the equation q(ρ)π(ρ) = c.

trust equilibrium that is stable in the replicator dynamic exists if and only ifxmin < x(ρ) <

xmax.

To prove the theorem, we equate p(ρ) in (3) with p∗ in (11) and solve for x, which gives

xmax = min

[dκ(1 − τ)

c + d(1 − τ),

2bκ(1 − τ)√a2 + 4ab− (a + 2bτ)

](17)

as the maximum feasible network size for the stage game. Now (16) shows that for suf-ficiently small c > 0, equilibrium profits are strictly positive in the variable parochialismgame as well. To determine xmin, we first find ρmin, the level of parochialism such thatexpected payoffs in a trust equilibrium are zero (by setting (16) equal to zero and solvingfor ρ) and then letting xmin = x(ρmax). We illustrate this situation in Fig. 3.

Note that it will generally be the case that xmin and xmax differ across totally orderedsubsets of ascriptive filters. The reason is that one subset may implement large size withlittle heterogeneity (thus allowing a larger xmax as the information costs of larger size wouldbe partially offset by lesser communication difficulty) while another subset may implementa high level of heterogeneity even for relatively small size. Similarly, if equilibrium payoffsdiffer across networks (because of different parameters a, b, c, d, τ, k and c), xmin will alsovary.

6. Parochialism and the anonymous market

In our model, individuals not in networks make up the anonymous pool of traders,unconditionally defecting and receiving a payoff normalized to zero. We now study apopulation-level equilibrium in which agents may migrate among one or more networksand the anonymous pool, and they do so when movement would increase their expectedpayoffs. For simplicity, we assume the cost of movement is zero and that networks acceptall immigrants who satisfy the network’s ascriptive filter. We will identify the conditionsunder which parochial networks may survive under these conditions.


It is easy to see that if ascriptive filters are very coarse such that, given some populationcomposition, a limited range of sizes of groups is possible, it may be that no feasible networkfalls within the xmin and xmax identified in Theorem 6. Our first condition for the presenceof parochial network is that at least one filter be sufficiently fine in the following sense.Since there are n traits, there are 2n possible trait profiles a ∈ A. Let f(a) be the frequencyof profile a ∈ A in the population, and let F(a) be the fraction of the population that satisfiesascriptive filter a ∈ A. We thus have

F(a) =∑b≥a

f(b).

For simplicity, we suppose the cost to an agent of moving from the anonymous pool to anetwork that will accept him is zero, and networks accept all immigrants who satisfy thenetwork’s ascriptive filter. On any totally ordered subset Po ⊂ P, we can associate thedegree of parochialism ρ with a particular ascriptive filter a(ρ) ∈ A, in which case we havex(ρ) = F(a(ρ)). Moreover, if Po is a maximal totally ordered subset, F(a(0)) = 1 andF(a(n)) = 0.14 We call a maximal totally ordered set Po ⊂ P ε-fine if, for every filteraN of a network N ∈ Po, there is a networkM ∈ Po with associated filter aM such that|F(aN) − F(aM)| < ε. Let X be the size of the population. We then have the followingtheorem.

Theorem 7. Suppose there is anε-fine maximal totally ordered subsetPo ⊂ P, whereδ = (xmax −xmin)/X for xmax and xmin given byTheorem 6. Then there exists an ascriptivefilter a and a networkN using ascriptive filter a that is a stable equilibrium of the replicatordynamic.

Note that in equilibrium, all members of the population who satisfy the ascriptive filtera migrate from the anonymous pool to the network N.

Under the conditions given in Theorem 7, there must exist a filter and a level of parochial-ism implementing a group size within the given range, which by Theorem 6 supports a stableequilibrium with an average payoff π > 0. If there are no other networks, the network inquestion will attract all population members conforming to the filter.

Suppose now that agents can move not only to networks costlessly, but can also moveamong networks costlessly. We define a population-level equilibrium as a set of networksN1, . . . ,Nk such that (a) each network is a stable equilibrium of the replicator dynamic; (b)no individual can gain by moving from the anonymous pool to a network; (c) no individualcan gain by moving from one network to another network; and (d) there is no ascriptivefilter a such that a network based on a could draw individuals from either one of the existingnetworks or from the anonymous pool. We have the following theorem.

Theorem 8. Suppose the conditions ofTheorem 7 are satisfied. Then there is a population-level equilibrium with at least one network.

14 This assumes that no agent has all n traits, which will be the case, for instance, if the traits include nationalorigin.


To prove this theorem, let N1 be the viable network with highest payoff. We know sucha network exists by Theorem 7 and the fact that there are only a finite number of possiblenetworks. If (a)–(d) in the paragraph preceding the theorem are satisfied, we are done. Ifnot, (d) must be violated. No additional network can draw members from N1, since thelatter has the highest possible payoff. Among the viable networks that can draw membersfrom the anonymous pool, let N2 be the one with the highest payoff. If (a)–(d) are nowsatisfied, we are done. Otherwise only (d) can be violated. We repeat the process until (d)is no longer violated.

7. Conclusion

Networks have properties that allow them to persist in a market economy despite theirrelative inability to exploit economies of scale and the other efficiency-enhancing propertiesof markets. Among these properties, and the one explored in this paper, is the capacity ofnetworks to support enforcement of prosocial behavior among network members. Networkshave this capacity by virtue of their ability to reduce information costs, thus permitting theemergence of ‘ trusting’ Nash equilibria that do not exist, or are unstable, when informationcosts are high. Our particular model of these relationships could readily be extended to cap-ture other salient aspects of the determinants of network formation, parochial exclusion, andnetwork extinction. For example, because parochialism makes networks not only smaller,but more homogeneous as well, corresponding efficiency-enhancing effects of similarity orsocial affinity with parochial networks may be important.

The value of the informal contractual enforcement capacities of networks, the viability ofnetworks, the range of viable network sizes, and the range of feasible degrees of parochialismall depend importantly on the nature of the goods and services that make up economicexchanges. Kollock (1994, p. 341) investigated “ the structural origins of trust in a system ofexchange” using an experimental design based on the exchange of goods of variable quality.He found that trust in and commitment to trading partners as well as a concern for one’sown and others’ reputations emerges when product quality is variable and non-contractiblebut not when it is contractible. These experimental results appear to capture some of thestructure of actual exchanges. Siamwalla’s (1978) study of marketing structures in Thailandcontrasts the impersonal structure of the wholesale rice market, where the quality of theproduct is readily assayed by the buyer, with the personalized exchange based on trust in theraw rubber market, where quality is impossible to determine at the point of purchase. Thus,were technologies to evolve such that quality and quantity of the goods being transacted arereadily subject to complete contracting, preferential trading within networks would be oflittle benefit and would likely be extinguished due to the implied foregone gains from trade.Conversely, were the economy to evolve in ways that heighten the problem of incompletecontracting we would expect to see growing economic importance of networks.

Applying this reasoning to our model, we consider the latter more likely. As productionshifts from goods to services, and within services to information-related services (Quah,1996), and as team-based production methods increase in importance, the gains from coop-eration will increase as well because such activities involve relatively high monitoring costsand are subject to costly forms of opportunism. If this is the case the benefits associated


with the mutual defect payoff (c) relative to the mutual cooperate payoff (b) will declineover time.15 Further, advances in communications technology arguably increase the num-ber (κ) of acquaintances from whom we can gather information at limited cost as well asincrease the intelligibility of messages, especially across cultural, linguistic, or other groupboundaries. The result would be to enhance the signal quality (p) for a given degree ofparochialism and hence level of network heterogeneity. The following are consequences.

The following are consequences. First, differentiating (13) we find that an increase in b, κor τ, or a decrease in c, raises the payoff π∗ to network members in a trust equilibrium for agiven size of network, thus making network membership more attractive relative to tradingin the anonymous pool. Second, differentiating (17) with respect to the same variables wefind that the effects of κ, τ, b, c, and xmax are of the same sign as for π∗. Thus, bettercommunication or higher payoffs to mutual cooperation relative to mutual defection in thenetwork will increase the largest network size at which the signal quality will support a trustequilibrium. Third, because π∗(x(ρ), ρ) is increasing in x at xmin, the upward shift in theπ∗ function reduces xmin. The intuition behind this result is that the increase in equilibriumpayoffs in the trust equilibrium for a given size allows a network to bear increased costs offorgone trading opportunities occasioned by smaller size without becoming unviable.

As a result potential networks characterized by ascriptive filters that would in the pasthave resulted in groups either too large or too small to sustain a trust equilibrium maybecome viable as contracts become more costly to express completely and to enforce and ascommunication improves, and the payoffs to existing parochial networks may rise relativeto the payoffs among anonymous traders.

On the other hand the kinds of social exclusion motivating network-based parochialismoften violate strongly held universalistic norms and may encounter legal prohibition or otherpublic policies motivated by a positive valuation of both tolerance and diversity of socialinteractions.

Acknowledgements

We are grateful to Katherine Baird, Roland Bénabou, Robert Boyd, Colin Camerer, Jef-frey Carpenter, Vincent Crawford, Steven Durlauf, Marcus Feldman, Edward Glaeser, AvnerGreif, David Laibson, Michael Macy, Paul Malherbe, Jane Mansbridge, Corinna Noelke,Paul Romer, Barkley Rosser, Leigh Tesfatsion, Martin Weitzman, Peyton Young, partici-pants in seminars at the NBER Summer Institute, The Santa Fe Institute, The University of

15 An increase in the cooperative payoff b does not make the standard prisoner’s dilemma interaction any ‘easierto solve’ of course, but it may enhance evolutionary pressures for the emergence of new rules of interaction thateffectively mitigate the dilemma. Wade (1988, p. 774–775) describes such a process:

a significant number of the villages (in one small part of Upland South India) have institutions for the provisionof public goods and services, which are autonomous of outside agencies in origin and operation. . . .Only a fewmiles may separate a village with a substantial amount of corporate organization from others with none. . . Whythe differences between villages? It is not because of differences in norms or values, for the villages are locatedwithin a small enough area for the culture to be uniform. It is rather because of differences in net collectivebenefit.


Siena, and Yale University for perceptive comments, and to the John D. and Catherine T.MacArthur Foundation and The Santa Fe Institute for financial support.

Appendix A. Proofs

Proof of Theorem 3. At the marketplace for good i, the number ξi of buyers and the numberχi of sellers are independently distributed binomial random variables with means xiψi(ρ)and xi(1 − ψi(ρ)), respectively, and variance xiψi(ρ)(1 − ψi(ρ))/2. The expected numberof agents not finding a trade is thus E[|ξi −χi|], where the expectation is with respect to theproduct distribution. Consider a single marketplace, and let ψ = ψi(ρ) be the probabilityan agent is a buyer, given level of parochialism ρ. Let x be a random variable that takes thevalue 1 with probability ψ and −1 with probability 1−ψ. The sum of x independent randomvariables distributed according to x has expected value x(2ψ− 1) and variance 4xp(1 −ψ).We assume x(ρ) large enough relative to k that the normal approximation to the binomial issufficiently accurate (x > 10 is enough to ensure this). The excess number of buyers is thusdistributed as a normal variate with mean x(2ψ − 1) and variance 4xp(1 − ψ). It is easy tocheck that the probability of obtaining a trade is given by

q(x, ψ) = 1 − g(x, ψ)

x,

where

g(x, ψ) = 4 e−(1−2ψ)2x/8ψ(1−ψ)

√ψ(1 − ψ)x

2π+ (2ψ − 1)x erf

[(2ψ − 1)x√8ψ(1 − ψ)x

]

and erf[y] = (2/√π)∫ y

0 e−t2 dt. We then find that

∂q

∂x= e−(1−2ψ)2x/8ψ(1−ψ)ψ(1 − ψ)√

πψ(1 − ψ)x/2(x − g(x, ψ)),

which is strictly positive. Thus q(x,ψ) is increasing in x. Since x = x(ρ) is a decreasingfunction of ρ, q is a decreasing function of ρ via its first argument.

The expression for ∂q/∂ψ is complicated, but has the opposite sign of

2 e(1−2ψ)2x/8ψ(1−ψ) erf

[(2ψ − 1)x√8ψ(1 − ψ)x

]+ (1 − 2ψ)

√2√

πψ(1 − ψ)x.

By expanding the integral in the erf function, we can show that this expression has thesame sign as (2ψ − 1) for x ≥ 1. Thus if increased parochialism increases the averagedisparity between buyers and sellers in market i, q is a decreasing function of ρ via itssecond argument as well. �

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